HEREDITARY TORSION THEORIES AND CONNECTEDNESS, DISCONNECTEDNESS OF TOPOLOGICAL SPACES

Abstract

Using topological congruences, a Hoehnke radical for topological spaces can be defined as in universal algebra. For most of the well-known classes
of algebras, an ideal-hereditary Hoehnke radical (= hereditary torsion theory) always determines a corresponding pair of Kurosh-Amitsur radical and semisimple classes. Here it is shown that an ideal-hereditary Hoehnke radical of topological spaces need not determine a corresponding pair of Kurosh-Amitsur radical and semisimple classes (= connectednesses and disconnectednesses). In fact, it is shown that there are exactly five hereditary torsion theories of topological spaces of which two are not Kurosh-Amitsur radicals.